A Tensor Rank Theory, Full-Rank Tensors and Base Subtensors
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A matrix always has a full-rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. Furthermore, the rows (columns) of this matrix are linear combinations of the rows (columns) of this matrix, corresponding to that submatrix. These two properties form the corner stones of matrix theory. We may call that submatrix a base submatrix of this matrix, and the two properties the sub-full-rank property and the base submatrix property, respectively. In this paper, we explore these two properties for tensors. We first present a theory for tensor ranks such that they are natural extension of matrix ranks. We propose some axioms for tensor rank functions. Then we introduce proper and strongly proper tensor rank functions. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the concepts of full-rank tensors and base subtensors, and show that the max-Tucker tensor rank function always has the sub-full-rank property and the base subtensor property. We then define the closure for a proper tensor rank function, and show that the closure of a proper tensor rank function is a proper tensor rank function with the sub-full-rank property. We also show that the smallest tensor rank function is strongly proper and has the sub-full-rank property.
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